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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

170 Chapter 2 • The Derivative and Its Properties

Solution
(a) For c = 2,

2
2
f (x) = x − 5x and f (2) = 2 − 5 · 2 = −6
The rate of change of f at c = 2 is
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2
2
f (x) − f (2) (x − 5x) − (−6) x − 5x + 6
f (2) = lim = lim = lim
′
x→2 x − 2 x→2 x − 2 x→2 x − 2
(x − 2)(x − 3)
= lim = lim (x − 3) = −1
x→2 x − 2 x→2
2
(b) If c is any real number, then f (c) = c − 5c, and the rate of change of f at c is
2
2
2
2
f (x) − f (c) (x − 5x) − (c − 5c) (x − c ) − 5(x − c)
′
f (c) = lim = lim = lim
x→c x − c x→c x − c x→c x − c
(x − c)(x + c) − 5(x − c) (x − c)(x + c − 5)
= lim = lim
x→c x − c x→c x − c
= lim (x + c − 5) = 2c − 5
x→c
R
NOW WORK Problem 17 and AP Practice Problem 3.
EXAMPLE 3 Finding the Rate of Change in a Biology Experiment
In a metabolic experiment, the mass M of glucose decreases according to the function
M(t) = 4.5 − 0.03t 2
where M is measured in grams (g) and t is the time in hours (h). Find the reaction
rate M (t) at t = 1 h.
′
Solution
The reaction rate at t = 1 is M (1).
′
2
M(t) − M(1) (4.5 − 0.03t ) − (4.5 − 0.03)
′
M (1) = lim = lim
t→1 t − 1 t→1 t − 1
2
2
−0.03t + 0.03 (−0.03)(t − 1) (−0.03)(t − 1)(t + 1)
= lim = lim = lim
t→1 t − 1 t→1 t − 1 t→1 t − 1
= −0.03 · 2 = −0.06
The reaction rate at t = 1 h is −0.06 g/h. That is, the mass M of glucose at t = 1 h is
changing at the rate of −0.06 g/h or decreasing at the rate of 0.06 g/h.
NOW WORK Problem 43.

3 Find Average Velocity and Instantaneous Velocity

Average velocity is a physical example of an average rate of change. For example,
s  f (t) consider an object moving along a horizontal line with the positive direction to the
0
right, or moving along a vertical line with the positive direction upward. The object’s
t
O location at time t = 0 is called its initial position. The initial position is usually marked
as the origin O on the line. See Figure 4. We assume the position s at time t of the
f (0) s  f (t) s
object from the origin is given by a function s = f (t). Here s is the signed, or directed,
distance (using some measure of distance such as centimeters, meters, feet, etc.) of
the object from O at time t (in seconds or hours). The function f is usually called
Figure 4 t is the travel time. s is the
signed distance of the object from the the position function of the object. Motion along a line is sometimes referred to as
origin at time t. rectilinear motion.

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